Geometry is the study of space in relation with distance, shape, size, and position of figures.

A circle is defined as a set of points equidistant from a central point, this distance is the circle's radius.

Trig Function Definitions

Solving Right Triangles

Circle

Distance

(x1-x0)²+(y1-x0)²=r² If the circle is centred at the origin (0, 0), then the equation simplifies to x²+y²=r²

double
distance(Point2d a, Point2d b)
{
	return sqrt((a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y));
}

Angle

To find angle, in radians, between two points:

double
angle(Point2d a, Point2d b)
{
	return atan2(b.y - a.y, b.x - a.x);
}

Position

To find a point on a circle:

Point2d
PtCircle2d(Point2d c, double r, double angle)
{
	return Pt2d(
		c.x + r * cos(angle),
		c.y + r * sin(angle));
}

Parallel

The slope of the first line is m1=(y²−y¹)/(x²−x¹) and the slope of the second is m2=(y⁴−y³)/(x⁴−x³). The lines are parallel if and only if m1=m2.

Let'?'s say we are given the center point of the circle and its radius. We can now create a loop which iterates from Center.x-Radius to Center.x+Radius or maybe even downwards from Center.x+Radius to Center.x-Radius. Now we have one point on the radius which is the center of the circle and one point which we have the X to, which is located on the circumference. We can then calculate the Y position of this point using the distance formula as in:

Radius = Sqrt ((P1.x - P2.x) ^2 + (P1.y - P2.y) ^2)

cos(x) = 1 - (x^2/2!) + (x^4/4!)...  (An even function)
sin(x) = x -(x^3/3!) + (x^5/5!)....  (An odd function)

add both series together but keep all signs positive and you have

e^x = 1 + x+ (x^2/2!) + (x^3/3!).....

So e^ipi + 1 =0

Degree Minute Position to Decimal Position

d = M.m / 60
Decimal Degrees = Degrees + .d

Example:
To convert 124° 44.740, a DMS coordinate, to DD.

44.740(m.m) / 60 = 0.74566667
124(degrees) + 0.74566667(.d) = 124.0.74566667

And so 124° 44.740 is 124.0.74566667 in Decimal Degrees.

• Paul Bourke
• Pi Is Wrong

Fractals are infinitely complex patterns that are self-similar across different scales.

mandelbrot.c

void
mandel(Uint32 *dst)
{
	int width = 640, height = 480, max = 254;
	int row, col;
	for(row = 0; row < height; row++) {
		for(col = 0; col < width; col++) {
			double c_re = (col - width / 1.5) * 4.0 / width;
			double c_im = (row - height / 2.0) * 4.0 / width;
			double x = 0, y = 0;
			Uint32 iteration = 0;
			while(x * x + y * y <= 4 && iteration < max) {
				double x_new = x * x - y * y + c_re;
				y = 2 * x * y + c_im;
				x = x_new;
				iteration++;
			}
			putpixel(dst, col, row, (iteration % 2) * 0xFFFFFF);
		}
	}
}

Mandelbrot without fixed point

See the complete SDL2 source.

mandel(-2.0 * NORM_FACT, -1.2 * NORM_FACT, 0.7 * NORM_FACT, 1.2 * NORM_FACT);

typedef unsigned char Uint8;
typedef signed char Sint8;
typedef unsigned short Uint16;
typedef signed short Sint16;

#define NORM_BITS 8
#define NORM_FACT ((Sint16)1 << NORM_BITS)

Uint16 WIDTH = 600;
Uint16 HEIGHT = 400;

int
iterate(Uint16 real0, Uint16 imag0)
{
	Uint8 i;
	Sint16 realq, imagq, real = real0, imag = imag0;
	for(i = 0; i < 255; i++) {
		realq = (real * real) >> NORM_BITS;
		imagq = (imag * imag) >> NORM_BITS;
		if((realq + imagq) > (Sint16)4 * NORM_FACT)
			break;
		imag = ((real * imag) >> (NORM_BITS - 1)) + imag0;
		real = realq - imagq + real0;
	}
	return i;
}

void
mandel(Sint16 realmin, Sint16 imagmin, Sint16 realmax, Sint16 imagmax)
{
	Uint16 x, y,
		deltareal = (realmax - realmin) / WIDTH,
		deltaimag = (imagmax - imagmin) / HEIGHT,
		real0 = realmin,
		imag0;
	for(x = 0; x < WIDTH; x++) {
		imag0 = imagmax;
		for(y = 0; y < HEIGHT; y++) {
			putpixel(pixels, x, y, iterate(real0, imag0));
			imag0 -= deltaimag;
		}
		real0 += deltareal;
	}
}

• Julia Set
• Mandelbrot Set
• Mandelbroth By Hand